The Blackman-Harris (BH) window family is a straightforward
generalization of the Hamming family introduced in
§3.2. Recall from that discussion that the
generalized Hamming family was constructed using a summation of three
shifted and scaled aliased-sinc-functions (shown in Fig.3.8). The
Blackman-Harris family is obtained by adding still more shifted sinc
functions:

(4.26)

where
, and
is the length zero-phase rectangular window (nonzero for
).
The corresponding window transform is given by

(4.27)

where
denotes the
rectangular-window transform, and
as usual.

Note that for
, we obtain the rectangular window, and for
,
the BH family specializes to the generalized Hamming family.

Relative to the generalized Hamming family (§3.2), we
have added one more cosine weighted by
. We now therefore
have three degrees of freedom to work with instead of two. In
the Hamming family, we used one degree of freedom to normalize the
window amplitude and the second was used either to maximize roll-off
rate (Hann) or side-lobe rejection (Hamming). Now we can use two
remaining degrees of freedom (after normalization) to optimize these
objectives, or we can use one for each, resulting in three subtypes
within the Blackman window family.

The classic Blackman window of the previous section is a three-term
window in the Blackman-Harris family (
), in which one degree of
freedom is used to minimize side-lobe level, and the other is used to
maximize roll-off rate. Harris [101, p. 64] defines the
three-term
Blackman-Harris window as the one which uses both degrees of
freedom to minimize side-lobe level. An improved design is given in
Nuttall [196, p. 89], and its properties are as follows:

Side lobes roll off
per octave in the absence of aliasing
(like rectangular and Hamming)

All degrees of freedom (scaling aside) are used to minimize side
lobes (like Hamming)

Figure 3.14 plots the three-term Blackman-Harris Window and its
transform. Figure 3.15 shows the same display for a much
longer window of the same type, to illustrate its similarity to the
rectangular window (and Hamming window) at high frequencies.

Note that the frequency-domain implementation of the Hann window requires
no multiplies in linear fixed-point data formats [188].

Similarly, any Blackman window may be implemented as a 5-point
smoother in the frequency domain. More generally, any
-term
Blackman-Harris window requires convolution of the critically sampled
spectrum with a smoother of length
.